862 research outputs found
Recommended from our members
Forward Limit Sets of Semigroups of Substitutions and Arithmetic Progressions in Automatic Sequences
This thesis deals with symbolic sequences generated by semigroups of substitutions acting on finite alphabets.
First, we investigate the underlying structure of certain automatic sequences by studying the maximum length A(d) of the monochromatic arithmetic progressions of difference d appearing in these sequences. For example, for the Thue-Morse sequence and a class of generalised Thue-Morse sequences, we give exact values of A(d) or upper bounds on it, for certain differences d. For aperiodic, primitive, bijective substitutions and spin substitutions, which are generalisations of the Thue-Morse and Rudin-Shapiro substitutions, respectively, we study the asymptotic growth rate of A(d). In particular, we prove that there exists a subsequence (d_n) of differences along which A(d_n) grows at least polynomially in d_n. Explicit upper and lower bounds for the growth exponent can be derived from a finite group associated to the substitution considered.
Next, we introduce the forward limit set Î of a semigroup S generated by a family of substitutions of a finite alphabet, which typically coincides with the set of all possible s-adic limits of that family. We provide several alternative characterisations of the forward limit set. For instance, we prove that Î is the unique maximal closed and strongly S-invariant subset of the space of all infinite words, and we prove that it is the closure of the image under S of the set of all fixed points of S. It is usually difficult to compute a forward limit set explicitly; however, we show that, provided certain assumptions hold, Î is uncountable, and we supply upper bounds on its size in terms of logarithmic Hausdorff dimension
Anomalous Reflection from Phase Gradient Metasurfaces for Arbitrary Incident Angles
Dissertation (MEng (Electronic Engineering))--University of Pretoria, 2023.Over the past few decades radar cross section (RCS) manipulation has become increasingly important. This increase in interest is due to the development and improvement of stealth technology. While many RCS manipulation techniques exist in the literature, most of these display certain shortcomings. The main disadvantages being complex target designs and narrow frequency bandwidth effectiveness. Metasurfaces are used to address these faults effectively for an array of practical applications. Checkerboard metasurfaces consists of an array of artificial magnetic conductor (AMC) elements, specifically two distinct AMC elements with phase differences of 180âŠ. This causes phase cancellation between the AMC elements and redirects the scattered energy away from the angle of incidence. The other RCS manipulating metasurface is the phase gradient metasurface (PGM). This study will focus on predicting the reflected wave directions from PGMs with various phase gradients
for an arbitrary incident wave. The prediction of the reflected wave direction from PGMs are currently restricted to perpendicular incidence or small angles close to the normal vector.
The reflected wave directions from PGMs are determined in the literature by utilising the generalised Snellâs law of reflection. This method is restricted by the relationship of the incident angle and phase gradient magnitude. If the critical value is exceeded the scattered wave direction becomes a complex value. Negative reflection was introduced to the adapted Snellâs law to ensure the predicted reflected wave direction values remain real. However, it is shown that additional energy is also observed close to the plane of the PGM which is not predicted by any of the predicted modes. Array theory is used to determine the scan angle of an antenna array. The PGM can also be viewed as an antenna array where each AMC represents an antenna element with a magnitude and phase value. This study shows that the predicted scattered wave direction is accurately estimated by combining array theory concepts with the adapted Snellâs law.
The proposed method of prediction is compared to a variety of simulated and measured metasurfaces. The reflected wave directions for a dual gradient metasurface with various incident angles are simulated in a computational electromagnetic (CEM) software package, CST Studio Suite, and compared to the proposed prediction method. A single gradient metasurface is designed at a different frequency and its bistatic and monostatic RCS is measured in the Compact Antenna Test Range (CATR) at the University of Pretoria.Electrical, Electronic and Computer EngineeringMEng (Electronic Engineering)Unrestricte
Beltrami fields exhibit knots and chaos almost surely
In this paper, we show that, with probability
, a random Beltrami field exhibits chaotic regions that coexist with invariant tori of complicated topologies. The motivation to consider this question, which arises in the study of stationary Euler flows in dimension 3, is V.I. Arnoldâs 1965 speculation that a typical Beltrami field exhibits the same complexity as the restriction to an energy hypersurface of a generic Hamiltonian system with two degrees of freedom. The proof hinges on the obtention of asymptotic bounds for the number of horseshoes, zeros and knotted invariant tori and periodic trajectories that a Gaussian random Beltrami field exhibits, which we obtain through a nontrivial extension of the NazarovâSodin theory for Gaussian random monochromatic waves and the application of different tools from the theory of dynamical systems, including KolmogorovâArnoldâMoser (KAM) theory, Melnikov analysis and hyperbolicity. Our results hold both in the case of Beltrami fields on
and of high-frequency Beltrami fields on the 3-torus
Growth and integrability of some birational maps in dimension three
Motivated by the study of the Kahan--Hirota--Kimura discretisation of the
Euler top, we characterise the growth and integrability properties of a
collection of elements in the Cremona group of a complex projective 3-space
using techniques from algebraic geometry. This collection consists of maps
obtained by composing the standard Cremona transformation
with projectivities that permute
the fixed points of and the points over which
performs a divisorial contraction. More specifically, we show that three
behaviour are possible: (A) integrable with quadratic degree growth and two
invariants, (B) periodic with two-periodic degree sequences and more than two
invariants, and (C) non-integrable with submaximal degree growth and one
invariant.Comment: 46 pages, 6 figures, 7 tables, comments are welcom
A survey of the homology cobordism group
In this survey, we present most recent highlights from the study of the
homology cobordism group, with a particular emphasis on its long-standing and
rich history in the context of smooth manifolds. Further, we list various
results on its algebraic structure and discuss its crucial role in the
development of low-dimensional topology. Also, we share a series of open
problems about the behavior of homology -spheres and the structure of
. Finally, we briefly discuss the knot concordance group
and the rational homology cobordism group ,
focusing on their algebraic structures, relating them to ,
and highlighting several open problems. The appendix is a compilation of
several constructions and presentations of homology -spheres introduced by
Brieskorn, Dehn, Gordon, Seifert, Siebenmann, and Waldhausen.Comment: 31 pages; 26 theorems (connecting several results) and 26 open
problems (with different levels of difficulty); to appear in the Bulletin of
the American Mathematical Societ
Fundamental Investigation on Polishing of Internal Structures Made by Laser-based Powder Bed Fusion
This study aims at utilizing a self-developed hybrid polishing system to establish polishing capabilities of electropolishing (EP), abrasive fluid polishing (AFP), multiple polishing in different sequences and innovative hybrid polishing for the internal structures prepared by laser-based powder bed fusion (L-PBF). By studying polishing effects on various inner surfaces, the relationships between polishing processes and material removal mechanisms of L-PBF surface features are established considering microstructural differences of surface features. The thesis starts with a comprehensive introduction of the project and then a literature review outlining L-PBF process, application of typical L-PBF internal structures, surface characteristics, advantages and limitations of current polishing methods for L-PBF inner surfaces. Breakthroughs in the surface finish of L-PBF internal structures are required in order to improve surface quality to meet the specific requirements of product performance. It is notified that EP and AFP exhibit complementary advantages in the polishing of internal structures among various polishing technologies. In the chapter 3, different types of inner surfaces for fundamental investigation and typical internal structures for application development are designed, and prepared by L-PBF using 316L stainless steel and Ti6Al4V powders. Then, un-sintered powders and sintered area are characterized as common surface features on L-PBF top, face up, side and face down surfaces considering the differences of morphology and microstructure. Meanwhile, an innovative hybrid polishing system which could carry out EP, AFP, their multiple polishing and hybrid polishing is established for the surface improvement of the L-PBF inner surfaces and internal structures. In the chapter 4, a polishing mode consisting of two-step EP is developed by using different potentials and polishing time for L-PBF 316L stainless steel and Ti6Al4V inner surfaces after parametric study in the developed polishing system. Based on material removal characteristics of surface features, the effectiveness and high efficiency of the two-step EP are demonstrated. Considering polishing characteristics of AFP, the inlet design of polishing chamber is improved and the material removal process of various L-PBF internal surfaces are discussed by analyzing the evolution of surface morphology, roughness and microstructure on cross sections in the chapter 5. In the chapter 6, multiple polishing in different sequences and hybrid polishing based on EP and AFP are investigated for L-PBF inner surfaces. It is found that L-PBF inner surfaces after single polishing can be further improved by multiple polishing in different sequences because of the complementary characteristics of EP and AFP in removing L-PBF surface features. In addition, hybrid polishing can perform EP and AFP simultaneously without interfering with each other, showing great potential in improving polishing efficiency of L-PBF inner surfaces. In the chapter 7, different polishing processes are applied to three typical L-PBF internal structures. The superiority of multiple polishing in different sequences and hybrid polishing for L-PBF inner structures is verified. Overall, it is confirmed that the self-developed innovative hybrid polishing system is capable of polishing various L-PBF internal structures with reliable results. Fundamental research on material removal characteristics of L-PBF surface features during polishing provides a theoretical basis for the applications of multiple and hybrid polishing based on EP and AFP. High efficiency and cost- effectiveness make the hybrid polishing system a strong candidate for industrial implementation in the surface finish of L-PBF internal structures
On the effectiveness of neural priors in modeling dynamical systems
Modelling dynamical systems is an integral component for understanding the
natural world. To this end, neural networks are becoming an increasingly
popular candidate owing to their ability to learn complex functions from large
amounts of data. Despite this recent progress, there has not been an adequate
discussion on the architectural regularization that neural networks offer when
learning such systems, hindering their efficient usage. In this paper, we
initiate a discussion in this direction using coordinate networks as a test
bed. We interpret dynamical systems and coordinate networks from a signal
processing lens, and show that simple coordinate networks with few layers can
be used to solve multiple problems in modelling dynamical systems, without any
explicit regularizers
- âŠ